398 XV NORMED ALGEBRAS AND SPECTRAL THEORY For each function /e ^rc(Sp(A/)), we have f(Nn) =f(N) \ En. For this is immediately seen to be true when/is a polynomial in C and I (15.11.1.1); for /e ^c(Sp(N))? the result then follows by the Stone-Weierstrass theorem (7.3.1) and (15.11.1.3); and the general case then follows by (15.11.2). We shall see that the study of N reduces to that of the Nn. Consider for example the eigenvalues of N: it is clear that x is an eigenvector of N if and only if each of its projections xn on En is either zero or an eigenvector of Nn, corresponding to an eigenvalue independent of n. We have then the following characterization: (15.11.6) For aeSp(Nn) to be an eigenvalue of Nn, it is necessary and sufficient that fj,n ({a}) ^ 0. The corresponding eigenvectors are those belonging to the image ofEn under the projector (P{^(NH)} and this image is a one-dimen- sional subfpace. Let a be an eigenvalue of Nn, and let F be the kernel of od En — Nn. Then by hypothesis F is closed and nonzero. Also it is stable under TV*, because NH - (N* • x) = N* * (Nn • x) = ocN* • x. Hence F is stable with respect to the algebra generated by Nn and N*. If we identify Nn with multiplication by the class of lc in L^(Sp(Nn), /*„), then F is identified with a subspace of the type Lc(Sp(Nn), <pM • nn), where M is a universally measurable set such that /zw(M) 7* 0 (15.10.6). We assert that the support of the measure v = (pM - jurt consists of a single point. If not, there would exist two disjoint closed subsets B, C of Sp(Nn), such that v(B n M) ^ 0 and v(C n M) ^ 0. The function C<PB n Ni(C) (resp. £q>c n M(Q) would be equal almost everywhere (with respect to v) to a<pBnM(Q (resp. a(pCnM(C))- In other words, we should have C = a almost everywhere in B n M and almost everywhere in C n M, which is absurd because B n C = 0. Hence we may assume that M = {/?}, and then it is clear that Lc(Sp(JVn), (pu - un) is of dimension 1, and that the restriction of Nn to this space is the homothety with ratio fi. If a e Sp(JV) is an eigenvalue of N, then the set J of integers n such that a is an eigenvalue of Nn is not empty. If Dn is the one-dimensional subspace of En generated by the eigenvectors of Nn corresponding to the eigenvalue a, then the eigenspace E(a; N) of N corresponding to a (11.1) is the Hilbert sum of the Dn for neJ. If J is finite, the number of elements of J (which is equal to the dimension of E(a; N)) is called the multiplicity of a. If J is infinite, then a is said to have infinite multiplicity. It is easily checked that this notion of multiplicity agrees with the notion of the multiplicity of a representation introduced in (15.10.9). Also it follows from the remarks above that if a, /? are distinct eigenvalues of N, then the eigenspaces E(a; N) and E(/?; N) are orthogonal.), such that T(gm) • a( tends to 0 and T(gm) • a'n tends to